1. Bayesian Learning in ME and HME Architectures - This Specific Aim will be to develop and apply Markov Chain Monte Carlo methodology to two specific types of neural networks: Mixtures-of-Experts (ME) and Hierarchical Mixtures-of-Experts (HME) Architectures. Recently, Peng, Jacobs and Tanner (1994) developed a Bayesian learning scheme for such architectures. The overall goal will be to further study and extend this methodology. The first specific subaim is to extend and investigate the ME and HME architectures in the binary response case. The second subaim will be to extend this ME and HME methodology to handle censored response data. The third subaim will be to compare this ME and HME methodology with competing nonparametric methods such as CART, MARS, generalized additive models (GAM's), and projection pursuit regression (PPR). The fourth specific subaim will be to develop methods to prune the ME and HME architectures. 2. Strategies for Computing the Marginal Likelihood - While algorithms such as data augmentation, the Gibbs sampler, the Metropolis algorithm and the Metropolis-Hastings algorithm have facilitated computations regarding estimation and prediction, the problem of computing the marginal likelihood remains an open problem. Such Markov Chain Monte Carlo algorithms yield a sample from the posterior- the marginal likelihood (or marginal density of the data) is obtained by integrating the likelihood function with respect to the prior density. The first subaim will be to develop and assess a new approach to this problem. The second subaim will be to validate this methodology using real data sets. The third subaim will be to compare this approach to those developed by other researchers. 3. Development of Software to Accompany Methodology - This Specific Aim is to develop transportable, documented, efficient code to. support the methodology developed under this grant. In particular, software will be written to apply the mixtures-of-experts and hierarchical mixtures-of- experts architectures to regression problems, binary response problems, categorical response problems and censored regression problems. Code will also be written to evaluate the marginal density of the sample data, thus allowing the comparison of competing models.